Optimal. Leaf size=158 \[ \frac{(c+d x) \tanh \left (\frac{e}{2}+\frac{f x}{2}+\frac{i \pi }{4}\right )}{3 a^2 f}+\frac{(c+d x) \tanh \left (\frac{e}{2}+\frac{f x}{2}+\frac{i \pi }{4}\right ) \text{sech}^2\left (\frac{e}{2}+\frac{f x}{2}+\frac{i \pi }{4}\right )}{6 a^2 f}+\frac{d \text{sech}^2\left (\frac{e}{2}+\frac{f x}{2}+\frac{i \pi }{4}\right )}{6 a^2 f^2}-\frac{2 d \log \left (\cosh \left (\frac{e}{2}+\frac{f x}{2}+\frac{i \pi }{4}\right )\right )}{3 a^2 f^2} \]
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Rubi [A] time = 0.109139, antiderivative size = 158, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {3318, 4185, 4184, 3475} \[ \frac{(c+d x) \tanh \left (\frac{e}{2}+\frac{f x}{2}+\frac{i \pi }{4}\right )}{3 a^2 f}+\frac{(c+d x) \tanh \left (\frac{e}{2}+\frac{f x}{2}+\frac{i \pi }{4}\right ) \text{sech}^2\left (\frac{e}{2}+\frac{f x}{2}+\frac{i \pi }{4}\right )}{6 a^2 f}+\frac{d \text{sech}^2\left (\frac{e}{2}+\frac{f x}{2}+\frac{i \pi }{4}\right )}{6 a^2 f^2}-\frac{2 d \log \left (\cosh \left (\frac{e}{2}+\frac{f x}{2}+\frac{i \pi }{4}\right )\right )}{3 a^2 f^2} \]
Antiderivative was successfully verified.
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Rule 3318
Rule 4185
Rule 4184
Rule 3475
Rubi steps
\begin{align*} \int \frac{c+d x}{(a+i a \sinh (e+f x))^2} \, dx &=\frac{\int (c+d x) \csc ^4\left (\frac{1}{2} \left (i e+\frac{\pi }{2}\right )+\frac{i f x}{2}\right ) \, dx}{4 a^2}\\ &=\frac{d \text{sech}^2\left (\frac{e}{2}+\frac{i \pi }{4}+\frac{f x}{2}\right )}{6 a^2 f^2}+\frac{(c+d x) \text{sech}^2\left (\frac{e}{2}+\frac{i \pi }{4}+\frac{f x}{2}\right ) \tanh \left (\frac{e}{2}+\frac{i \pi }{4}+\frac{f x}{2}\right )}{6 a^2 f}-\frac{\int (c+d x) \text{csch}^2\left (\frac{e}{2}-\frac{i \pi }{4}+\frac{f x}{2}\right ) \, dx}{6 a^2}\\ &=\frac{d \text{sech}^2\left (\frac{e}{2}+\frac{i \pi }{4}+\frac{f x}{2}\right )}{6 a^2 f^2}+\frac{(c+d x) \tanh \left (\frac{e}{2}+\frac{i \pi }{4}+\frac{f x}{2}\right )}{3 a^2 f}+\frac{(c+d x) \text{sech}^2\left (\frac{e}{2}+\frac{i \pi }{4}+\frac{f x}{2}\right ) \tanh \left (\frac{e}{2}+\frac{i \pi }{4}+\frac{f x}{2}\right )}{6 a^2 f}-\frac{d \int \coth \left (\frac{e}{2}-\frac{i \pi }{4}+\frac{f x}{2}\right ) \, dx}{3 a^2 f}\\ &=-\frac{2 d \log \left (\cosh \left (\frac{e}{2}+\frac{i \pi }{4}+\frac{f x}{2}\right )\right )}{3 a^2 f^2}+\frac{d \text{sech}^2\left (\frac{e}{2}+\frac{i \pi }{4}+\frac{f x}{2}\right )}{6 a^2 f^2}+\frac{(c+d x) \tanh \left (\frac{e}{2}+\frac{i \pi }{4}+\frac{f x}{2}\right )}{3 a^2 f}+\frac{(c+d x) \text{sech}^2\left (\frac{e}{2}+\frac{i \pi }{4}+\frac{f x}{2}\right ) \tanh \left (\frac{e}{2}+\frac{i \pi }{4}+\frac{f x}{2}\right )}{6 a^2 f}\\ \end{align*}
Mathematica [A] time = 1.08094, size = 241, normalized size = 1.53 \[ \frac{\left (\sinh \left (\frac{1}{2} (e+f x)\right )-i \cosh \left (\frac{1}{2} (e+f x)\right )\right ) \left (\cosh \left (\frac{3}{2} (e+f x)\right ) \left (2 c f+2 d \tan ^{-1}\left (\tanh \left (\frac{1}{2} (e+f x)\right )\right )-i d \log (\cosh (e+f x))-d e+d f x\right )+2 i \sinh \left (\frac{1}{2} (e+f x)\right ) \left (-3 c f-4 d \tan ^{-1}\left (\tanh \left (\frac{1}{2} (e+f x)\right )\right )+2 i d \log (\cosh (e+f x))+d \cosh (e+f x) \left (-2 \tan ^{-1}\left (\tanh \left (\frac{1}{2} (e+f x)\right )\right )+i \log (\cosh (e+f x))+e+f x\right )+2 d e-d f x-i d\right )+d \cosh \left (\frac{1}{2} (e+f x)\right ) \left (-6 \tan ^{-1}\left (\tanh \left (\frac{1}{2} (e+f x)\right )\right )+3 i \log (\cosh (e+f x))+3 e+3 f x-2 i\right )\right )}{6 a^2 f^2 (\sinh (e+f x)-i)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.103, size = 113, normalized size = 0.7 \begin{align*}{\frac{2\,dx}{3\,{a}^{2}f}}+{\frac{2\,de}{3\,{f}^{2}{a}^{2}}}-{\frac{{\frac{2\,i}{3}} \left ( 3\,ifdx{{\rm e}^{fx+e}}+3\,ifc{{\rm e}^{fx+e}}-id{{\rm e}^{fx+e}}+dfx+d{{\rm e}^{2\,fx+2\,e}}+cf \right ) }{ \left ({{\rm e}^{fx+e}}-i \right ) ^{3}{f}^{2}{a}^{2}}}-{\frac{2\,d\ln \left ({{\rm e}^{fx+e}}-i \right ) }{3\,{f}^{2}{a}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.19648, size = 346, normalized size = 2.19 \begin{align*} \frac{1}{3} \, d{\left (\frac{3 \,{\left (2 \, f x e^{\left (3 \, f x + 3 \, e\right )} +{\left (-6 i \, f x e^{\left (2 \, e\right )} - 2 i \, e^{\left (2 \, e\right )}\right )} e^{\left (2 \, f x\right )} - 2 \, e^{\left (f x + e\right )}\right )}}{3 \, a^{2} f^{2} e^{\left (3 \, f x + 3 \, e\right )} - 9 i \, a^{2} f^{2} e^{\left (2 \, f x + 2 \, e\right )} - 9 \, a^{2} f^{2} e^{\left (f x + e\right )} + 3 i \, a^{2} f^{2}} - \frac{2 \, \log \left (-i \,{\left (i \, e^{\left (f x + e\right )} + 1\right )} e^{\left (-e\right )}\right )}{a^{2} f^{2}}\right )} + c{\left (\frac{6 \, e^{\left (-f x - e\right )}}{{\left (9 \, a^{2} e^{\left (-f x - e\right )} - 9 i \, a^{2} e^{\left (-2 \, f x - 2 \, e\right )} - 3 \, a^{2} e^{\left (-3 \, f x - 3 \, e\right )} + 3 i \, a^{2}\right )} f} + \frac{2 i}{{\left (9 \, a^{2} e^{\left (-f x - e\right )} - 9 i \, a^{2} e^{\left (-2 \, f x - 2 \, e\right )} - 3 \, a^{2} e^{\left (-3 \, f x - 3 \, e\right )} + 3 i \, a^{2}\right )} f}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.53945, size = 398, normalized size = 2.52 \begin{align*} \frac{2 \, d f x e^{\left (3 \, f x + 3 \, e\right )} - 2 i \, c f +{\left (-6 i \, d f x - 2 i \, d\right )} e^{\left (2 \, f x + 2 \, e\right )} + 2 \,{\left (3 \, c f - d\right )} e^{\left (f x + e\right )} -{\left (2 \, d e^{\left (3 \, f x + 3 \, e\right )} - 6 i \, d e^{\left (2 \, f x + 2 \, e\right )} - 6 \, d e^{\left (f x + e\right )} + 2 i \, d\right )} \log \left (e^{\left (f x + e\right )} - i\right )}{3 \, a^{2} f^{2} e^{\left (3 \, f x + 3 \, e\right )} - 9 i \, a^{2} f^{2} e^{\left (2 \, f x + 2 \, e\right )} - 9 \, a^{2} f^{2} e^{\left (f x + e\right )} + 3 i \, a^{2} f^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 8.02081, size = 173, normalized size = 1.09 \begin{align*} \frac{\frac{2 i d e^{e} e^{- 2 f x}}{3 a^{2} f^{2}} - \frac{2 i c e^{3 e} + 2 i d x e^{3 e}}{3 a^{2} f} - \frac{\left (6 c f e^{2 e} + 6 d f x e^{2 e} + 2 d e^{2 e}\right ) e^{- f x}}{3 a^{2} f^{2}}}{- i e^{3 e} - 3 e^{2 e} e^{- f x} + 3 i e^{e} e^{- 2 f x} + e^{- 3 f x}} - \frac{2 d x}{3 a^{2} f} - \frac{2 d \log{\left (i e^{e} + e^{- f x} \right )}}{3 a^{2} f^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.32547, size = 285, normalized size = 1.8 \begin{align*} \frac{2 \, d f x e^{\left (3 \, f x + 3 \, e\right )} - 6 i \, d f x e^{\left (2 \, f x + 2 \, e\right )} + 6 \, c f e^{\left (f x + e\right )} - 2 \, d e^{\left (3 \, f x + 3 \, e\right )} \log \left (e^{\left (f x + e\right )} - i\right ) + 6 i \, d e^{\left (2 \, f x + 2 \, e\right )} \log \left (e^{\left (f x + e\right )} - i\right ) + 6 \, d e^{\left (f x + e\right )} \log \left (e^{\left (f x + e\right )} - i\right ) - 2 i \, c f - 2 i \, d e^{\left (2 \, f x + 2 \, e\right )} - 2 \, d e^{\left (f x + e\right )} - 2 i \, d \log \left (e^{\left (f x + e\right )} - i\right )}{3 \, a^{2} f^{2} e^{\left (3 \, f x + 3 \, e\right )} - 9 i \, a^{2} f^{2} e^{\left (2 \, f x + 2 \, e\right )} - 9 \, a^{2} f^{2} e^{\left (f x + e\right )} + 3 i \, a^{2} f^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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